 Research article
 Open Access
Interactive visualization of multidimensional coincidence spectra
 Miroslav Morháč^{1}Email author and
 Vladislav Matoušek^{1}
 Received: 22 June 2007
 Accepted: 23 November 2007
 Published: 23 November 2007
Abstract
The paper presents direct visualization techniques of multidimensional nuclear spectra as well as visualization techniques based on projections of embedded subspaces. While the first group of graphical models is limited to four dimensions, the second one can be theoretically extended to any dimension. The presented algorithms of visualization have been implemented in nuclear data acquisition, processing and visualization system developed at the Institute of Physics, Slovak Academy of Sciences. The paper focuses on presentation of nuclear spectra. However the majority of algorithms can be successfully applied for visualization of scalar arrays of other data types.
PACS Code: 29.85.+c, 07.05.Rm
Keywords
 Visualization Technique
 Display Mode
 Coincidence Spectrum
 Visualization Algorithm
 Nuclear Spectrum
1. Introduction
The power of computers to collect, store and manipulate experimental data has increased dramatically. In today's nuclear physics experiments the number of detectors being included in the measurements is going up to one hundred or more. The results of such measurements, however, generate such large data sets as to be nearly incomprehensible. Scanning these large sets of numbers to determine trends and relationships is a tedious and ineffective process. To address this problem the physicists have turned to visualization of experimental data. If the data are converted to a visual form, the trends are often immediately apparent. Without visualization much of the increased power of computers would be wasted because experiments are poor at gaining insight from data presented in numerical form.
The goal of visualization of experimental data is an improved understanding of the result of the information gathered during experiment. It is one of the most powerful and direct ways how the huge amount of information can be conveyed in a form comprehensible to a human eye. As a result, development of visualization algorithms takes on great significance, offering a promising technology for transforming an indigestible mass of numbers into a medium, which experimenters can understand, interpret and explore.
The visualization techniques presented in this work make it possible to display either raw experimental spectra, processed data or to make slices of the same or lower dimensionality in an interactive way. They allow obtaining an imagination about event distribution and correlations in coincidence spectra up to fivedimensional space.
The objective of the paper is to present visualization techniques and graphical models to display multidimensional nuclear spectra (histograms), which were implemented in the nuclear data acquisition, processing and visualization system [1, 2]. Though the software package is designed mainly for use with nuclear data, any kind of data can be processed as well. Other systems for nuclear spectra graphical representation were presented in [3, 4].
2. Direct visualization techniques of scalar fields
A scalar variable is a single quantity, in the case of nuclear spectra – counts, which can be represented as a function of independent variables – particle energies. Most scalar visualization techniques use a consistent approach across one, two, or threedimensional fields. More recent techniques, e.g. of the visualization of threedimensional fields, attempt to show the full threedimensional variations of a scalar variable within a volume field. These techniques include isosurfaces, particle clouds, volume slicing and sampling planes [5–7].
The sophisticated visualization algorithms are presented in [8]. The paper presents conventional as well as newly developed visualization techniques and graphical models. The structure and complexity of the algorithms lend themselves for implementation in online live mode during the data acquisition or processing. The pictures can be simultaneously updated.
One can select various attributes of the display, e.g. color of the spectrum, the limits of the displayed part of the spectrum, window, marker, type of scale, and various display modes, slices, to rotate two, or moredimensional data. In the abovementioned paper, we have developed the direct visualization algorithms up to fourdimensional data.
2.1. Twodimensional spectra
where t_{ xx }, t_{ xy }, t_{ yx }, t_{ yy }, t_{ yz }, v_{ x }, v_{ y }are transform coefficients reflecting translation in both original twodimensional scalar field (in x, y dimensions as well as in counts) and in the position on screen, scaling, rotation around zaxis and elevation of the view. The position of a point on the screen is x', y' and
n_{ x }, n_{ y }are numbers of nodes of a regular grid. The model proposed in such a way allows:
• to choose and display any part of the spectrum by setting x_{ min }, x_{ max }, y_{ min }, y_{ max }to appropriate values
• to set any range of displayed counts – c_{ min }, c_{ max }
• to place the display of spectrum anywhere on the screen
• to rotate and elevate the view of the spectrum
• to change the density of display nodes. This is important when displaying accumulated spectra in online mode, i.e., during the acquisition of spectra.
2.2. Threedimensional spectra
where t_{ xx }, t_{ xy }, t_{ xz }, t_{ yx }, t_{ yy }, t_{ yz }, v_{ x }, v_{ y }, v_{ z }, are display transform coefficients reflecting translations in both original threedimensional scalar field (in x, y, z dimensions as well as in counts) and in the position on screen, scaling, rotation around axes x, y, z and
x = x_{ min }+ k_{ x }·i; y = y_{ min }+ k_{ y }·j; z = z_{ min }+ k_{ z }·k
n_{ x }, n_{ y }, n_{ z }are numbers of nodes of regular grid. The model proposed in such a way allows:

to choose and display any part of threedimensional array – x_{ min }, x_{ max }, y_{ min }, y_{ max }, z_{ min }, z_{ max }

to choose any range of displayed counts – c_{ min }, c_{ max }

to place the picture anywhere on the screen

to rotate the spectrum around the axes x, y, z

to change the density of display nodes.
2.3. Fourdimensional spectra
Now the counts is a function of four parameters (particle energies), i.e., c = f(x, y, z, v). Let us imagine that instead of one channel belonging to one point of 3D space in threeparameter nuclear spectrum visualization now this point represents a slice in the fourth parameter, i.e.,
c_{i, j, k}(v) = f(x_{ i }, y_{ j }, z_{ k }, v)
3. Technique of successive projections of embedded subspaces
The dimensionality of abovepresented visualization techniques is limited to four. However, with increasing dimensionality of nuclear spectra the requirements in developing of multidimensional scalar visualization techniques becomes striking. In principle, the abovementioned algorithms can be used even for higher dimensions by employing a new technique of embedded subspaces. Using this technique we divide the multidimensional space into outer subspace and one or more successive inner (embedded) subspaces, all of dimensionalities more convenient to human imagination.
The goal is to propose a technique that allows one to localize and scan interesting parts (peaks) in multidimensional spectra. Moreover it should permit to find correlations in the data, mainly among neighboring points, and thus to discover prevailing trends around multidimensional peaks.
The proposed technique makes benefit of specific character and features of nuclear spectra. It utilizes the fact that the interesting objects (peaks) have shape of quasi Gaussians. Further, in enormous multidimensional space the events are distributed very sparsely, which allows to preserve main features of data even after reducing the dimensionality by employing projection functional. Successive decreasing the dimensionality makes it possible to determine the positions of appropriate multidimensional peaks.
Without loss of generality, we shall assume the reduction of the space up to twodimensional one. Other alternatives are also possible, but the display of twodimensional array using perpendicular view allows utilizing screen area the most efficiently. Let us start with threedimensional spectrum f(x, y, z). Let us apply a projection functional reducing dimensionality by one to twodimensional array, e.g.
f^{(1)}(x, y) = F[f(x, y, z)].
or maximum in a slice
f^{(1)}(x, y) = max{f(x, y, z)},
where z ∈ <z_{ min }, z_{ max }>
or any other suitable operation. Let us display each channel i, j in the form of a mark with size proportional to f^{(1)}(i, j). Again, because of the most efficient way of utilizing the screen, in place of the mark we choose a rectangle. The rectangle represents a "window" into the subspace. Inside of the rectangle, we can display the slice f(i, j, z), z ∈ <z_{ min }, z_{ max }>. From the distribution of rectangles, one can find out the positions of interesting peaks, then focus the view to an appropriate region or to zoom a slice to full screen size, respectively.
Then inside of each rectangle belonging to the channel i_{1}, i_{2} we display twodimensional slice f(i_{1}, i_{2}, x_{3}, x_{4}), using any of the twodimensional above presented graphical models.
In the first case in each rectangle window belonging to the channel i_{1}, i_{2} one can display threedimensional slice f(i_{1}, i_{2}, x_{3}, x_{4}, x_{5}) using any of the threedimensional graphical models. In the second one, in each rectangle belonging to the channel i_{1}, i_{2} one can display twodimensional distribution of f^{(1)}(i_{1}, i_{2}, x_{3}, x_{4}) again in the form of rectangles. Then in each rectangle belonging to the channel i_{1}, i_{2}, i_{3}, i_{4}, one can display the onedimensional slice f(i_{1}, i_{2}, i_{3}, i_{4}, x_{5}). Employing this algorithm and using successive zooming one can localize the positions of fivedimensional peaks.
where j is the level of merging. Apparently, if p is odd 0level subspaces are onedimensional.
Obviously, from theoretical point of view this algorithm has no limitation. However, due to technical limitations of today's computers (sizes of memories, resolution of displays) the practical meaning of these formulas for higher values of p is rather restricted. Finally, we would like to emphasize that the given algorithm of embedded subspaces presents one of the possible approaches to cope with the problem of visualization of multidimensional nuclear spectra. In principle one can change dimensionality of subspaces at every level of merging, rotate subspaces, define other projection functional etc.
3.1. Threedimensional spectra
One can see simultaneously the distribution of the twodimensional projection (yellow squares) together with onedimensional slices. One can observe correlations among neighboring points inside of rectangles as well as correlation of corresponding points in the rectangles in both x and y directions.
3.2. Fourdimensional spectra
3.3. Fivedimensional spectra
4. Conclusion
In the paper we have presented conventional as well as new developed visualization algorithms of nuclear spectra. For 3D spectra we have proposed particle gradient display technique and isosurface display technique. Raw data can be interpolated using Bspline algorithms up to 4th degree. For 4D spectra we have designed the algorithms based on slicing in fourth dimension, pies display mode as well as isovolume display mode.
Furthermore, we have derived new technique of visualization of multidimensional spectra based on projections of embedded subspaces. This allows one, in interactive way, to localize interesting parts in the data of this kind, to find correlations among neighboring points and to discover trends in multidimensional data.
The visualization algorithms presented have been implemented in the data acquisition, processing and visualization system DaqProVis which is being developed at Institute of Physics, Slovak Academy of Sciences [9]. The algorithms for the display of 2D spectra have also been implemented in ROOT system in TSpectrum2Painter class (SPECTRUMPAINTER directory) [10]. In near future we plan to implement in ROOT also the visualization algorithms for 3D spectra.
Declarations
Acknowledgements
The work is supported by the Grant Agency of Slovak Republic through contract GAV 2/7117/27.
Authors’ Affiliations
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